Re: isprime() runtime jump between 10^18 and 10^20?

On 2026-01-06 11:59, Aurel Page wrote:
> Dear Hermann,
> 
> There are two prime testing functions in pari:
> - ispseudoprime: a fast pseudo-primality test, which in particular
> guarantees primality of numbers up to 2^64.
> - isprime: a certified primality test, which only runs ispseudoprime
> for numbers up to 2^64, but performs a much more costly test
> otherwise.
> In addition, as you know you should expect operations on integers that
> do not fit in a long to be much slower, as we immediately switch to
> multiprecision integers (t_INT).
> I did a quick test on my laptop (don't consider these numbers as 
> universal):
> - ispseudoprime was about 17 times slower for numbers between 2^63 and
> 2^64 than for numbers between 2^64 and 2^65 (long -> t_INT).
> - isprime was about 12 times slower than ispseudoprime for numbers
> between 2^64 and 2^65.
> Together, these make a jump of x200 for isprime from [2^63,2^64] to 
> [2^64,2^65].
> 
> Note that 2^64 ~ 10^19, corresponding to your observed jump.
> 
Thanks for the explanation. But long is a signed type, shouldn't the 
transition from long to t_INT happen at +2^63?


I don't know why I did not think on parallel execution of GP script 
before, now I did. And because of the results I changed sum() in new 
oeis sequence to parsum() because that works for single as well as 
pthread gp engine.

I found the 7 parallel "par*" GP statements and searched for their use 
on oeis.org. No hit for 5, one (false) hit for "parfor" (Maple code) and 
only two hits for "parsum".


Runtime reduced from 5min to 5 seconds, and from 23h to 19 minutes using 
parsum()!

hermann@x3950-X6:~$ nproc
384
hermann@x3950-X6:~$ numactl -C 0-191 gp -q
? a(n)=parsum(k=1, 10^n, isprime(k^2+(k+1)^2));
? #
    timer = 1 (on)
? a(9)
cpu time = 14min, 51,150 ms, real time = 4,819 ms.
68588950
? a(10)
cpu time = 59h, 57min, 42,370 ms, real time = 18min, 46,398 ms.
614983774
?


In addition I did the parsum() count for 10^10 < k <= 2*10^10, and now 
the 19 minutes with lots of ispseudoprime() for processing 10^10 ks 
became 1h (not too bad) for processing the same amount of ks with zero 
ispseudoprime():

? parsum(k=10^10+1,2*10^10,isprime(k^2+(k+1)^2));
cpu time = 191h, 45min, 10,976 ms, real time = 1h, 3,472 ms.
?


In my C++ code I was responsible to make sure that OpenMP creates work 
on all cores. Using the  "par*" GP functions is so much easier, thanks 
to the dev team!


Regards,

Hermann.